On the global convergence of a new spectral residual algorithm for nonlinear systems of equations
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On the global convergence of a new spectral residual algorithm for nonlinear systems of equations Alessandra Papini1 · Margherita Porcelli2
· Cristina Sgattoni3
Received: 6 June 2020 / Accepted: 4 November 2020 © The Author(s) 2020
Abstract We present a derivative-free method for solving systems of nonlinear equations that belongs to the class of spectral residual methods. We will show that by endowing a previous version of the algorithm with a suitable new linesearch strategy, standard global convergence results can be attained under mild general assumptions. The robustness of the new method is therefore potentially improved with respect to the previous version as shown by the reported numerical experiments. Keywords Nonlinear systems of equations · Spectral residual methods · Global convergence · Nonmonotone linesearch
1 Introduction In this work we propose a variant of the derivative-free spectral residual method PandSR presented in [16], for solving nonlinear systems of equations of the form: F(x) = 0,
(1)
with the aim of obtaining stronger global convergence results when F : Rn → Rn is a continuously differentiable mapping. Indeed, the sequence generated by Pand- SR was
B
Margherita Porcelli [email protected] Alessandra Papini [email protected] Cristina Sgattoni [email protected]
1
Dipartimento di Ingegneria Industriale, Università degli Studi di Firenze, viale G.B. Morgagni 40, 50134 Florence, Italy
2
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
3
Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, viale G.B. Morgagni 67a, 50134 Florence, Italy
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proved to be convergent under mild standard assumptions, but only in a more specific setting it was shown in [16] that the limit point is also a solution of (1). Inspired by [11], we adopt here a different linesearch strategy, which allows us to obtain a more general and nontrivial result for methods that do not make any use of derivatives of f , and in fact was not established in [16]. Namely we can prove that at every limit point x ∗ of the sequence {xk } generated by the new algorithm, either F(x ∗ ) = 0 or the gradient of the merit function 1 f (x) = F(x)22 (2) 2 is orthogonal to the residual F:
∇ f (x ∗ ), F(x ∗ ) = J (x ∗ )T F(x ∗ ), F(x ∗ ) = 0,
(3)
being J the Jacobian of F.1 Clearly the orthogonality condition (3) does not generally imply F(x ∗ ) = 0; however this result can be recovered under additional conditions, e.g. when J (x ∗ ) is positive (negative) definite. We further remark that the improvement with respect to Pand- SR is not only theoretical; as discussed in Sect. 4, the performed numerical experiments show that the new linesearch has a positive impact also on the practical behaviour of the method. Given the current iterate xk , spectral residual methods are methods of linesearch type which produce a new iterate xk+1 of the form: xk+1 = xk ± λk βk F(xk ) – both the residual ve
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